As far as I'm aware this is only an issue with hard tail guitars? (I may be wrong)

But basically, after chugging/trying to play really tight chords and stopping a chord dead, a high pitch noise is heard right after. I've heard of a device called a Jimmy Clip some people use but I don't think they're sold anymore. Is there any other solution to this?

This of course is only an issue live, in a recording I can chop it out easy.

Serious answer:
Try lowering your bridge pup a little, if that is the pup your using when your playing and having this problem. Otherwise lower your neck pup a little if that's the one your using when having this problem.

If you use thicker strings and don't have your nut cut out to the right size, the string could be sitting more on top of the nut than in the slot. I heard this as a rumor once; I don't know if it's true.

On the other hand, I had the same problem and found that putting a hairband around the strings behind the nut stopped the vibrations. They were ringing out behind the nut and the hairband mutes them. You could use something like electrical tape too.